Related papers: Differential equations in a tangent category I: Co…
We construct algorithms and topological invariants that allow us to distinguish the topological type of a surface, as well as functions and vector fields for their topological equivalence. In the first part (arXiv:2501.15657), we discused…
Interpreting tangency as a limit of two transverse intersections, we obtain a concrete formula to enumerate smooth degree $d$ plane curves tangent to a given line at multiple points with arbitrary order of tangency. Extending that idea, we…
Tate objects have been studied by many authors. They allow us to deal with infinite dimensional spaces by identifying some more structure. In this article, we set up the theory of Tate objects in stable $(\infty,1)$-categories, while the…
Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a solution, explicit examples and the notion of…
In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $e^x$ from classical differential calculus. A differential exponential map is an endomorphism which…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
Exponential actions defined by vector configurations provide a universal framework for several constructions of holomorphic dynamics, non-Kaehler complex geometry, toric geometry and topology. These include leaf spaces of holomorphic…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a…
A critical transition for a system modelled by a concave quadratic scalar ordinary differential equation occurs when a small variation of the coefficients changes dramatically the dynamics, from the existence of an attractor-repeller pair…
This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as…
A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations,…
Principal bundles have at least three different definitions, depending on the category of geometric objects studied. In Differential Geometry, they are defined as locally trivial projection map of smooth manifolds with an atlas whose…
Differential equation discovery, a machine learning subfield, is used to develop interpretable models, particularly in nature-related applications. By expertly incorporating the general parametric form of the equation of motion and…
In industry, shape optimization problems are of utter importance when designing structures such as aircraft, automobiles and turbines. For many of these applications, the structure changes over time, with a prescribed or non-prescribed…
New concept of conditional differential invariant is discussed that would allow description of equations invariant with respect to an operator under a certain condition. Example of conditional invariants of the projective operator is…
We study a system of partial differential equations with integer and fractional derivatives arising in the study of forced oscillatory motion of a viscoelastic rod. We propose a new approach considering a quotient of relations appearing in…
This paper deals with a complete invariant $R_X$ for cyclic quotient surface singularities. This invariant appears in the Riemann Roch and Numerical Adjunction Formulas for normal surface singularities. Our goal is to give an explicit…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
A categorical framework for modeling and analyzing systems in a broad sense is proposed. These systems should be thought of as `machines' with inputs and outputs, carrying some sort of signal that occurs through some notion of time. Special…